How to Round Any CSP Prasad Raghavendra University of Washington, Seattle David Steurer, Princeton University (In Principle)

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Presentation on theme: "How to Round Any CSP Prasad Raghavendra University of Washington, Seattle David Steurer, Princeton University (In Principle)"— Presentation transcript:

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How to Round Any CSP Prasad Raghavendra University of Washington, Seattle David Steurer, Princeton University (In Principle)

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Constraint Satisfaction Problem A Classic Example : Max-3-SAT Given a 3-SAT formula, Find an assignment to the variables that satisfies the maximum number of clauses. Equivalently the largest fraction of clauses

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Strong and Weak STRENGTH For every clause Á in = -local distributions ¹ Á over assignments to the variables of Á Vector variables v i,a within a clause Á satisfy all valid constraints (like triangle inequality) – the inner products are in the integral hull. WEAKNESS The above hard constraint is only for variables that participate together in a clause

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STEP 1 : Dimension Reduction Project the SDP solution along d =1/ Є 4 random directions. STEP 3 : Discretization Pick an Є –net for the d dimensional sphere Move every variable to the nearest point in the Є –net = finite = discretized instance STEP 2 : Throw away Discard clauses for which the corresponding inner products are not preserved within Є. = ‘ = New instance From STEP 2, We have discarded clauses for which inner products are not preserved within Є Discarding a clause P Forget about constraints corresponding to P Discretization changes inner product by Є For every remaining clause, all inner products are within 2Є of what it was.